Algebra Examples

$\frac{a^{2}}{a x - x^{2}} + \frac{x}{x - a}$

Step 1

Factor $x$ out of $a x - x^{2}$ .

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Step 1.1

Factor $x$ out of $a x$ .

$\frac{a^{2}}{x a - x^{2}} + \frac{x}{x - a}$

Step 1.2

Factor $x$ out of $- x^{2}$ .

$\frac{a^{2}}{x a + x (- x)} + \frac{x}{x - a}$

Step 1.3

Factor $x$ out of $x a + x (- x)$ .

$\frac{a^{2}}{x (a - x)} + \frac{x}{x - a}$

Step 2

Factor $- 1$ out of $x$ .

$\frac{a^{2}}{x (a - x)} + \frac{x}{- 1 (- x) - a}$

Step 3

Factor $- 1$ out of $- a$ .

$\frac{a^{2}}{x (a - x)} + \frac{x}{- 1 (- x) - (a)}$

Step 4

Factor $- 1$ out of $- 1 (- x) - (a)$ .

$\frac{a^{2}}{x (a - x)} + \frac{x}{- 1 (- x + a)}$

Step 5

Simplify the expression.

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Step 5.1

Move a negative from the denominator of $\frac{x}{- 1 (- x + a)}$ to the numerator.

$\frac{a^{2}}{x (a - x)} + \frac{- x}{- x + a}$

Step 5.2

Reorder terms.

$\frac{a^{2}}{x (a - x)} + \frac{- x}{a - x}$

Step 6

To write $\frac{- x}{a - x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{a^{2}}{x (a - x)} + \frac{- x}{a - x} \cdot \frac{x}{x}$

Step 7

Write each expression with a common denominator of $x (a - x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

Multiply $\frac{- x}{a - x}$ by $\frac{x}{x}$ .

$\frac{a^{2}}{x (a - x)} + \frac{- x \cdot x}{(a - x) x}$

Step 7.2

Reorder the factors of $(a - x) x$ .

$\frac{a^{2}}{x (a - x)} + \frac{- x \cdot x}{x (a - x)}$

Step 8

Combine the numerators over the common denominator.

$\frac{a^{2} - x \cdot x}{x (a - x)}$

Step 9

Simplify the numerator.

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Step 9.1

Rewrite $x \cdot x$ as $x^{2}$ .

$\frac{a^{2} - x^{2}}{x (a - x)}$

Step 9.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = x$ .

$\frac{(a + x) (a - x)}{x (a - x)}$

Step 10

Cancel the common factor of $a - x$ .

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Step 10.1

Cancel the common factor.

$\frac{(a + x) (a - x)}{x (a - x)}$

Step 10.2

Rewrite the expression.

$\frac{a + x}{x}$